By 1993 it was know that a quantum computer could perform certain
tasks asymptotically faster than any classical computer. Nonetheless
research into quantum computing was largely driven by academic
curiosity. In 1994 Peter Shor, a scientist working for Bell Labs,
devised a polynomial time algorithm for finding prime factors of large
numbers on a quantum computer. This discovery drew great attention to
the field of quantum computing.

Shor's algorithm is viewed as important because the difficulty of
finding prime factors of large numbers is relied upon for most
cryptography systems. If an efficient method of factoring large
numbers were to be discovered most of the current encryption schemes
would be easily compromised. While it has not been proven that
factoring large numbers can not be archived on a classical computer in
polynomial time, the fastest published algorithm for factoring large
number as of 2015 in
O(exp(n1/3(logn)2/3),
operations where n is the number of bits used to represent the
number: this runtime exceeds polynomial time. In contrast Shor's
algorithm runs in
O((log n)2*loglog n) on a quantum
computer, and then must perform O(log n) steps of post processing
on a classical computer. Overall this time is polynomial. This
discovery propelled the study of quantum computing forward, as such an
algorithm is much sought after. (Shor)